Strict positivity and a boundary contact

8 written marks

Let \(F_a(x)=(a+2)x^2-6x+a-1\), with real parameter \(a\).

1. Explain why \(a\gt -2\) is necessary if \(F_a(x)\gt 0\) for every real \(x\).

   [2]

2. Find all \(a\) for which \(F_a(x)\gt 0\) for every real \(x\).

   [4]

3. At \(\displaystyle a=\frac{-1+3\sqrt5}{2}\), find the x-value at which \(F_a(x)=0\).

   [2]

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